If stars are produced in the CMZ at a rate of \(\dot{M}_*\)  (in \({\rm M_\odot /yr}\) , assumed constant over the past few tens of millions of years) with a power-law initial mass functionWe normalize the IMF so that \(\int _{m_{min}}^{m_{max}} m (dn/dm)dm =\dot{M}_*\) . This normalization yields \(C = \dot{M}_*(s-2)/(m_{min}^{2-s}-m_{max}^{2-s})\) ,
so that \(dn/dm\)  itself has units of number of stars per \({\rm M_\odot }\)  per yr. (IMF) \(dn/dm = C m^{-s}\) , then the number of (massive) stars with masses between \(m\)  and \(m+dm\)  will be \(T(m)(dn/dm)dm\)  and their kinetic energy injection rate over the entire CMZ will be \(\dot{E}_{kin,1}T(m)(dn/dm)dm\) . Integrating over all stars with \(m>20\) , we obtain a total \(\dot{E}_{\rm kin, winds}\)  from such stars of \(\int _{m_{min}}^{m_{max}} \dot{E}_{kin,1}T(m)(dn/dm)dm\) .
Adopting standard parameters for a Salpeter IMF (\(s=2.35\) , \(m_{max}=120\) , and \(m_{min}=0.15\)  so that \(\langle m \rangle \sim 0.5\) ), and a \(\dot{M}_* \sim 0.1\)  (e.g. [1]} and references therein) we obtain \(\dot{E}_{\rm kin, winds} \sim 10^{40} {\rm erg/s}\)  for the entire volume of the CMZ to which the star formation estimate refers. Converting this to a per-volume estimate is not trivial. First, the star formation is not uniformly distributed over the CMZ volume, as is clear from [1]}, who compare different estimates of \(\dot{M}_*\)  in the CMZ using different tracers and studying different volumes: as can be seen in their Table 1, there is no strong correlation between the volume of the CMZ studied and the derived value of \(\dot{M}_*\) , which is to be expected if the star formation, massive stars, and molecular clouds are all spatially correlated and only cover a small fraction of the CMZ volume. If we assume that most of this star formation takes place in the same region of the central CMZ covered by the clouds studied in [3]} (radius of \(\sim 100\)  pc, height of \(\sim \pm 5\)  pc from the Galactic plane), then we obtain \(\dot{E}_{\rm kin, winds} \sim 4\times 10^{34}\, {\rm erg \, s^{-1} pc^{-3}}\)  (corresponding to a population of about  1500 O and B stars of all evolutionary stages within the CMZ). However, this estimate is very uncertain. It is most sensitive to our assumptions on the IMF and the volume within most of the star formation is confined. For example, if we assume a top-heavy IMF similar to that preferred for the central parsec (e.g. \(s=0.85\) , \(m_{min}=0.7\)  as in Model 8 of [4]}), we obtain \(\dot{E}_{\rm kin, winds} \sim 6\times 10^{35}\, {\rm erg \, s^{-1} pc^{-3}}\) . Conversely, for every factor-two increase in the Galactrocentric radius of the CMZ zone, where most of the star formation takes place, the estimate of \(\dot{E}_{\rm kin, winds}\)  is decreased by a factor of four. However, it is encouraging that, even with these large uncertainties, this first-principles estimate is of the same order of magnitude as the simple calculation of Eq. (REF ).
